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Theorem ltresr2 9841
 Description: Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
Assertion
Ref Expression
ltresr2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (1st𝐴) <R (1st𝐵)))

Proof of Theorem ltresr2
StepHypRef Expression
1 elreal2 9832 . . . 4 (𝐴 ∈ ℝ ↔ ((1st𝐴) ∈ R𝐴 = ⟨(1st𝐴), 0R⟩))
21simprbi 479 . . 3 (𝐴 ∈ ℝ → 𝐴 = ⟨(1st𝐴), 0R⟩)
3 elreal2 9832 . . . 4 (𝐵 ∈ ℝ ↔ ((1st𝐵) ∈ R𝐵 = ⟨(1st𝐵), 0R⟩))
43simprbi 479 . . 3 (𝐵 ∈ ℝ → 𝐵 = ⟨(1st𝐵), 0R⟩)
52, 4breqan12d 4599 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ⟨(1st𝐴), 0R⟩ < ⟨(1st𝐵), 0R⟩))
6 ltresr 9840 . 2 (⟨(1st𝐴), 0R⟩ < ⟨(1st𝐵), 0R⟩ ↔ (1st𝐴) <R (1st𝐵))
75, 6syl6bb 275 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (1st𝐴) <R (1st𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   class class class wbr 4583  ‘cfv 5804  1st c1st 7057  Rcnr 9566  0Rc0r 9567
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